# [R] Interpreting summary.lm for a 2 factor anova

David Winsemius dwinsemius at comcast.net
Fri Dec 2 21:03:25 CET 2016

```> On Dec 2, 2016, at 9:09 AM, David Winsemius <dwinsemius at comcast.net> wrote:
>
>>
>> On Dec 2, 2016, at 6:16 AM, Ashim Kapoor <ashimkapoor at gmail.com> wrote:
>>
>> Dear Pikal,
>>
>> All levels except the interactions are compared to the Intercept. I'm a
>> little confused as to what's going on in interaction terms eg. the cell
>> wool B : tension M. It's mean is :
>> 28.78 and 28.78 - 44.56 = -15.78 != 21.111.
>>
>> It's something like 44.56 (intercept) -16.333 (wool B) -.20.556 (tension
>> M)  + 21.111 (woolB:tensionM) = 28.782.
>>
>> I don't know how to sum up the above line in terms of differences
>> succinctly.
>
> The aov estimate will not exactly equal the observed mean (this is _statistics_ after all). You should be comparing the mean of that cell to the estimate:
>
> 44.556 + (-16.33) +(-20.556) + (21.11)

A respected participant advised me to look at this more closely. In this case (and I think in most such cases)  where there are the same number of parameters as there are means, the model is "saturated" and there is no difference:

with( warpbreaks, tapply( breaks, interaction(wool, tension), mean ) )
A.L      B.L      A.M      B.M      A.H      B.H
44.55556 28.22222 24.00000 28.77778 24.55556 18.77778

So the B:M estimate is identical up to rounding with the observed mean:

44.556 + (-16.33) +(-20.556) + (21.11)
[1] 28.78

>
> The difference between the observed mean and the estimated mean is known as a 'residual'

I've also been privately but gently chided for this misstatement. Residuals are the difference between data and estimates.

> and the squared sum of the all residuals is what this being minimized ... over all the cells including the one implicitly associated with the Intercept.
>
> This isn't really on-topic for Rhelp since you are not having difficulty in getting the R program to perform its duties, but are rather in need of statistical education. That not what this mailing list is set up for.
>
> --
> David.
>
>>
>>>
>>>> -----Original Message-----
>>>> From: R-help [mailto:r-help-bounces at r-project.org] On Behalf Of Ashim
>>>> Kapoor
>>>> Sent: Thursday, December 1, 2016 2:48 PM
>>>> To: r-help at r-project.org
>>>> Subject: [R] Interpreting summary.lm for a 2 factor anova
>>>>
>>>> Dear all,
>>>>
>>>> Here is a small example : -
>>>>
>>>>> model <- aov(breaks ~ wool * tension, data = warpbreaks)
>>>>> summary.lm(model)
>>>>
>>>> Call:
>>>> aov(formula = breaks ~ wool * tension, data = warpbreaks)
>>>>
>>>> Residuals:
>>>>    Min       1Q   Median       3Q      Max
>>>> -19.5556  -6.8889  -0.6667   7.1944  25.4444
>>>>
>>>> Coefficients:
>>>>              Estimate Std. Error t value Pr(>|t|)
>>>> (Intercept)      44.556      3.647  12.218 2.43e-16 ***
>>>> woolB           -16.333      5.157  -3.167 0.002677 **
>>>> tensionM        -20.556      5.157  -3.986 0.000228 ***
>>>> tensionH        -20.000      5.157  -3.878 0.000320 ***
>>>> woolB:tensionM   21.111      7.294   2.895 0.005698 **
>>>> woolB:tensionH   10.556      7.294   1.447 0.154327
>>>> ---
>>>> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
>>>>
>>>> Residual standard error: 10.94 on 48 degrees of freedom
>>>> Multiple R-squared:  0.3778,    Adjusted R-squared:  0.3129
>>>> F-statistic: 5.828 on 5 and 48 DF,  p-value: 0.0002772
>>>>
>>>>> model.tables(model,"e")
>>>> Tables of effects
>>>>
>>>> wool
>>>> wool
>>>>     A       B
>>>> 2.8889 -2.8889
>>>>
>>>> tension
>>>> tension
>>>>    L      M      H
>>>> 8.241 -1.759 -6.481
>>>>
>>>> wool:tension
>>>>   tension
>>>> wool L      M      H
>>>>  A  5.278 -5.278  0.000
>>>>  B -5.278  5.278  0.000
>>>>
>>>>
>>>>> model.tables(model,"m")
>>>> Tables of means
>>>> Grand mean
>>>>
>>>> 28.14815
>>>>
>>>> wool
>>>> wool
>>>>    A      B
>>>> 31.037 25.259
>>>>
>>>> tension
>>>> tension
>>>>   L     M     H
>>>> 36.39 26.39 21.67
>>>>
>>>> wool:tension
>>>>   tension
>>>> wool L     M     H
>>>>  A 44.56 24.00 24.56
>>>>  B 28.22 28.78 18.78
>>>>>
>>>>
>>>> I don't follow the output of summary.lm. I understand the output of
>>>> model.tables for effects and means. For instance what does 44.556
>>>> represent ? Is it the grand average ? The grand mean is 28.14815. Can
>>>> someone help me understand the output of summary.lm ?
>>>>
>>>> Best Regards,
>>>> Ashim
>>>>
>>>>     [[alternative HTML version deleted]]
>>>>
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>> 	[[alternative HTML version deleted]]
>>
>> ______________________________________________
>> R-help at r-project.org mailing list -- To UNSUBSCRIBE and more, see
>> https://stat.ethz.ch/mailman/listinfo/r-help
>> and provide commented, minimal, self-contained, reproducible code.
>
> David Winsemius
> Alameda, CA, USA
>
> ______________________________________________
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